Finite Group Databasebeta

← #14544030

Subgroups — #14544030

Order 2176.

440 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763780C212yesmaximal, complemented
2#11763653C212yesmaximal, complemented
3#11763652C212yesmaximal, complemented
4C2×C22⋊C4×D17C211yesmaximal, characteristic
5C24×D17C411yescharacteristic
6C22×C4×D17C411yescharacteristic
7C22×C4×D17C2211yescharacteristic
8C2×#63243—22no—
9C2×C22⋊C4×C17C411yescharacteristic
10C2×C22⋊Dic17C2211yescharacteristic
11C2×#63227C411yescharacteristic
12C2×#63227C2211yescharacteristic
13C22⋊C4×D17—22no—
14C22⋊C4×D17—22no—
15C22⋊#3624C2214yescomplemented
16#10587543—22no—
17C24×C17C2×C411yescharacteristic
18C23×D17—22no—
19C23×D17—21no—
20C23×D17—21no—
21C23×D17—22no—
22C23×D17—42no—
23C23×D17—42no—
24C23×D17C2×C411yescharacteristic
25C23×D17C2×C411yescharacteristic
26C22×C4×C17C2×C411yescharacteristic
27C22×C4×C17C2×C411yescharacteristic
28C22×Dic17C2×C411yescharacteristic
29C22×Dic17C2311yescharacteristic
30C2×C4×D17—21no—
31C2×C4×D17—21no—
32C2×C4×D17—42no—
33C2×C4×D17—42no—
34C2×C4×D17D412yes—
35C2×C4×D17D412yes—
36C2×#3624—24no—
37#63243—44no—
38C22⋊C4×C17—22no—
39C22⋊Dic17—22no—
40#63227—22no—
41#63227—22no—
42C23×C17—21no—
43C23×C17—21no—
44C23×C17—41no—
45C23×C17—41no—
46C23×C17M1612yescomplemented
47C23×C17C22×C411yescharacteristic
48C22×D17—21no—
49C22×D17—21no—
50C22×D17—22no—
51C22×D17—22no—
52C22×D17—24no—
53C22×D17—24no—
54C22×D17—21no—
55C22×D17—21no—
56C22×D17—48no—
57C22×D17—48no—
58C22×D17—42no—
59C22×D17—42no—
60C22×D17—42no—
61C22×D17—42no—
62C22×D17—41no—
63C22×D17—41no—
64C22×D17C22⋊C411yescharacteristic
65C22×D17C22⋊C411yescharacteristic
66C22×D17M1612yescomplemented
67C2×C4×C17—21no—
68C2×C4×C17—41no—
69C2×C4×C17—41no—
70C2×C4×C17C22⋊C412yes—
71C2×Dic17—21no—
72C2×Dic17—41no—
73C2×Dic17—41no—
74C2×Dic17C2×D412yes—
75C4×D17—24no—
76C4×D17—42no—
77C4×D17—42no—
78C4×D17—42no—
79#3624—44no—
80#1462—171nomaximal, Sylow
81C22×C17—21no—
82C22×C17—21no—
83C22×C17—22no—
84C22×C17—42no—
85C22×C17—42no—
86C22×C17—41no—
87C22×C17—41no—
88C22×C17C2×C22⋊C411yescharacteristic
89C22×C17C2×M1611yescharacteristic
90C22×C17C2×M1611yescharacteristic
91C2×D17—24no—
92C2×D17—24no—
93C2×D17—21no—
94C2×D17—21no—
95C2×D17—21no—
96C2×D17—21no—
97C2×D17—22no—
98C2×D17—48no—
99C2×D17—48no—
100C2×D17—42no—
101C2×D17—41no—
102C2×D17—41no—
103C2×D17—42no—
104C2×D17#9912yes—
105C2×D17#10012yes—
106C4×C17—22no—
107C4×C17—41no—
108Dic17—22no—
109Dic17—41no—
110C22×C22⋊C4—171no—
111#407—172no—
112#323—172no—
113#323—172no—
114C2×C17—21no—
115C2×C17—21no—
116C2×C17—42no—
117C2×C17#40711yescharacteristic
118C2×C17C2×#9911yescharacteristic
119C2×C17C2×#10011yescharacteristic
120D17—22no—
121D17—22no—
122D17—42no—
123C25—171no—
124C23×C4—171no—
125C23×C4—171no—
126C2×M16—342no—
127C2×C22⋊C4—171no—
128C2×C22⋊C4—171no—
129C2×C22⋊C4—171no—
130C2×C22⋊C4—342no—
131C2×C22⋊C4—342no—
132C2×C22⋊C4#59511yescharacteristic
133C22⋊C8—174no—
134C22⋊C8—342no—
135C17#146211yescharacteristic, complemented, Sylow
136C24—171no—
137C24—171no—
138C24—341no—
139C24—341no—
140C24—342no—
141C24—342no—
142C24—682no—
143C24—682no—
144C24C2×#59511yescharacteristic
145C22×C4—171no—
146C22×C4—172no—
147C22×C4—172no—
148C22×C4—171no—
149C22×C4—341no—
150C22×C4—341no—
151C22×C4—682no—
152C22×C4—682no—
153C22×C4C2×#59511yescharacteristic
154C22×C4C2×#59511yescharacteristic
155M16—684no—
156C2×C8—344no—
157C22⋊C4—22no—
158C22⋊C4—342no—
159C22⋊C4—342no—
160C22⋊C4—342no—
161C23—21no—
162C23—21no—
163C23—41no—
164C23—41no—
165C23—171no—
166C23—171no—
167C23—172no—
168C23—342no—
169C23—342no—
170C23—344no—
171C23—344no—
172C23—341no—
173C23—341no—
174C23—341no—
175C23—341no—
176C23—681no—
177C23—681no—
178C23—682no—
179C23—682no—
180C23—688no—
181C23—688no—
182C23—682no—
183C23—682no—
184C23#6324312yescomplemented
185C23C22×#59511yescharacteristic
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—172no—
190C2×C4—344no—
191C2×C4—341no—
192C2×C4—682no—
193C2×C4—682no—
194C2×C4—682no—
195C2×C4—681no—
196C2×C4—681no—
197C2×C4C22⋊#59512yes—
198C8—684no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—172no—
207C22—172no—
208C22—344no—
209C22—344no—
210C22—341no—
211C22—341no—
212C22—341no—
213C22—341no—
214C22—342no—
215C22—682no—
216C22—682no—
217C22—688no—
218C22—688no—
219C22—681no—
220C22—681no—
221C22C2×#6324311yescharacteristic
222C22D17⋊M1611yescharacteristic
223C22C2×C22⋊#59511yescharacteristic
224C4—22no—
225C4—41no—
226C4—342no—
227C4—681no—
228C2—21no—
229C2—21no—
230C2—42no—
231C2—342no—
232C2—342no—
233C2—682no—
234C2#1176378011yescharacteristic
235C2D17⋊#9911yescharacteristic
236C2D17⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C22⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC24×D17C4
cosocleC22×Dic17C23
hypercenterC2×C22⋊C4#595
hypocenterC17#1462
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11763780C2semidirect
2. #11763653C2semidirect
3. #11763652C2semidirect
15. C22⋊#3624C22semidirect
46. C23×C17M16semidirect
66. C22×D17M16semidirect
135. C17#1462semidirect
184. C23#63243semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117637802
#117636532
#117636522
C2×C22⋊C4×D172
#146217